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THE Reader is desired to refer to the Plate at end of book containing diagrams of the Zodiacal Planisphere, which has been spoken of in the Note in p. 99.

Fig. 1 is the Planisphere adjusted for the northern latitude of 30° 22' (where the longest day consists of fourteen equatorial hours), agreeably to the "Exemplification" given by Ptolemy in Chapter XV, Book 3. It represents that portion of the celestial sphere which is contained between the tropics: the central horizontal line is the equator; the curved line extending longitudinally from east to west is the ecliptic; the central perpendicular line is the meridian, or cusp of the 10th house; the other short lines, cutting the equator transversely, are the cusps of the other houses; that of the 1st house being the eastern horizon; that of the 7th, the western horizon. Hence, the distance from the 1st house to the meridian, or from the meridian to the 7th house, shows the semi-diurnal arc of any parallel of declination in the ecliptic; and the distance of the 7th house to the 4th, or from the 4th to the 1st, shows the semi-nocturnal arc. The distance from the cusp of one house to that of the next, taken on the same parallel, is also equal to two

temporal hours; thus, for instance, in the latitude above quoted, the semi-diurnal arc of 0° ♊ is 6 h. 50 m., or 102° 39' of the equator; consequently the diurnal temporal hour is equal to one equatorial hour and eight minutes, or to 17° 6' of the equator.

In his first example, Ptolemy directs 0° ♈ to be placed on the ascendant, so that the beginning of ♑ may be on the mid-heaven; 0° ♊ must, therefore, fall on the point A, distant from the mid-heaven 147° 44' of the equator, as measured by the line AB; because every point in the sphere always preserves one and the same parallel with the equator; and 6° ♊, in passing to the mid-heaven, must proceed along the line AB. In the present case, however, it is required to know how long 0° ♊ will be in coming to the ascendant, the given position of 0° ♈. Now 0° ♊ will be on the ascendant when it arrives at the point G; therefore the distance from A to C is the amount of the prorogation between 0° ♈ (when posited on the ascendant) and 0° ♊, and it is equal to 45° 5' of the equator. In the second example, 0° ♈ is placed on the mid-heaven, which position must be at D, so that 0° ♊ must necessarily be at E; and the distance from E to B, equal to 57° 44' of the equator, is the prorogation between 0° ♈ and 0° , when 0° ♈ is on the mid-heaven. In the third example, 0° ♈ supposed to be on the 7th house, descending, at F, so that ♋ is on the mid-heaven, and 0° ♊ at the point G, in advance of the mid-heaven 32° 16' of the equator, as shown by the distance BG. Now it is required to bring 0° ♊ to the 7th house (the place of 0° ♈), and it will be there on arriving at H, distant from B 102° 39' of the equator; but as 0° ♊ is already at G, the distance from G to H, equal to 70° 23' of the equator, is the amount of the prorogation between 0° ♈ and 0° ♊, when 0° ♈ is on the 7th house. The fourth example places 0° ♈ at I, three temporal hours past the meridian; 0° ♈ therefore falls on the point K, at the distance of 13 equatorial degrees before the meridian or mid-heaven, and will be three temporal hours past the meridian (the position of 0° ♊) on arriving at L, distant 51 equatorial degrees from the mid-heaven: the whole distance from K (the first position of 0° ♊) to L, its second position, equal to 64 degrees of the equator, is therefore the prorogation between 0° ♈ and 0° ♊, when 0° ♈ is past the meridian at the distance of three temporal hours. Ptolemy has also instanced two other positions for 0° ♈; viz. at two temporal hours past the meridian, and at two temporal hours before the occidental angle; or, in other words, on the cusp of the 9th house, and on that of the 8th. Now, if 0° ♈ be on the cusp of the 9th house, it must be at M, and 0° ♊ will be at N, distant 62 equatorial degrees from Q, which is also on the cusp of the 9th. If 0° ♈ be on the cusp of the 8th, it must be at O, and 0° ♊ will be at P, distant 66 equatorial degrees from R, which is also on the cusp of the 8th: these two several numbers of degrees will be the respective prorogations between 0° and 0° ♊, when 0° ♈ is placed on the 9th and 8th houses.

Ptolemy's "Exemplification" has been followed thus minutely in

order to show how perfectly Mr. Ranger's invention is adapted to assist (if not to supersede) arithmetical calculation; for, after the Planisphere has once been accurately laid down, a line drawn parallel to the equator, from the significator to the promittor, or to the promittor's pole of position, and measured by degrees of the equator, will accomplish the whole operation of ascertaining the amount of prorogation.

Fig. 2 is the Equator extended, *in plana*, on a scale proportionate to the planispheres in Figs. 1 and 3: it is divided into 360 degrees, and into equal time, as measured by the 24 hours of the earth's daily rotation on its axis, and by smaller portions of four minutes each, corresponding with degrees of the equator.

Fig. 3 is the Planisphere set for the latitude of Southern Britain, 51° 30' N., where the longest day is 16 h. 30 m., the semi-diurnal arc of 0° being consequently 7 h. 52 m., or 118° of the equator, and its diurnal temporal hour equal to one hour and nearly nineteen minutes of equatorial time, or to 19° 40' of the equator. In applying Ptolemy's examples, given in Chapter XV, Book 3, to this latitude, it will follow that, when 0° ♈ may be on the ascendant, 0° ♊ will be at A, and will subsequently arrive at the ascendant at C, after the passage of 29° 43' of the equator. When 0° ♈ may be on the mid-heaven at D, 0° ♊ will be at E, and will arrive at B, on the mid-heaven, after the passage of 57° 44' of the equator, as in Fig. 1. When 0° ♈ may be on the 7th house, at F, 0° ♊ will be at G, and will come to the 7th house, at H, after the passage of 85° 45' of the equator. If 0° ♈ be three temporal hours past the meridian, at I, 0° would be at K, again 13 equatorial degrees before the meridian, as in Fig. 1, and will be three temporal hours past the meridian, a position similar to that assumed for 0° ♈, on arriving at L, distant from the mid-heaven 59 equatorial degrees; thus making the whole distance, from K to L, 17 equatorial degrees. If 0° ♈ be on the 9th house, at M, 0° ♊ will be at N, distant from Q (also on the 9th house) about 67 equatorial degrees. If 0° ♈ be on the 8th house, at O, 0° ♊ will be at P, distant from R (also on the 8th house) about 76 equatorial degrees.

By taking the trouble to calculate the distances between the several positions given by Ptolemy, the Reader may satisfy himself of the sufficiency of this Planisphere for the purpose for which it was first projected; viz. for the more expeditious measurement of the arcs of direction. The Tables of Ascensions, extracted from the Almagest, in p. 152, will show that the arcs, as measured in Figs. 1 and 2 of the plate, exactly tally with the amounts of distance obtained by calculating arithmetically, according to the respective latitudes, as quoted in the Tables.

The slight view which has been here given of the Zodiacal Planisphere invented by Mr. Ranger, must not be considered as pretending to

offer a complete idea of its powers: they are so manifold and various, that another volume would be required to detail them fully; and it has now been used only in order to give a better illustration of Ptolemy's examples of the spaces of prorogation than mere words can do. To persons conversant with the mathematical part of astronomy, the facility with which a complete representation of zodiacal latitude, declination, the poles of position, crepusculine circles, and other phenomena, may be made by this Planisphere, will be sufficiently obvious from the accompanying Figures.

FINIS

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Figures 1-3